Year: 2020
Paper: 3
Question Number: 10
Course: UFM Mechanics
Section: Simple Harmonic Motion
No solution available for this problem.
In spite of the change to criteria for entering the paper, there was still a very healthy number of candidates, and the vast majority handled the protocols for the online testing very well. Just over half the candidates attempted exactly six questions, and whilst about 10% attempted a seventh, hardly any did more than seven. With 20% attempting five questions, and 10% attempting only four, overall, there were very few candidates not attempting the target number. There was a spread of popularity across the questions, with no question attracting more than 90% of candidates and only one less than 10%, but every question received a good number of attempts. Likewise, there was a spread of success on the questions, though every question attracted at least one perfect solution.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A light elastic spring $AB$, of natural length $a$ and modulus of elasticity $kmg$, hangs vertically with one end $A$ attached to a fixed point. A particle of mass $m$ is attached to the other end $B$. The particle is held at rest so that $AB > a$ and is released.
Find the equation of motion of the particle and deduce that the particle oscillates vertically.
If the period of oscillation is $\dfrac{2\pi}{\Omega}$, show that $kg = a\Omega^2$.
Suppose instead that the particle, still attached to $B$, lies on a horizontal platform which performs simple harmonic motion vertically with amplitude $b$ and period $\dfrac{2\pi}{\omega}$.
At the lowest point of its oscillation, the platform is a distance $h$ below $A$.
Let $x$ be the distance of the particle above the lowest point of the oscillation of the platform. When the particle is in contact with the platform, show that the upward force on the particle from the platform is
\[ mg + m\Omega^2(a + x - h) + m\omega^2(b - x). \]
Given that $\omega < \Omega$, show that, if the particle remains in contact with the platform throughout its motion,
\[ h \leqslant a\left(1 + \frac{1}{k}\right) + \frac{\omega^2 b}{\Omega^2}. \]
Find the corresponding inequality if $\omega > \Omega$.
Hence show that, if the particle remains in contact with the platform throughout its motion, it is necessary that
\[ h \leqslant a\left(1 + \frac{1}{k}\right) + b, \]
whatever the value of $\omega$.
This was the least popular question on the paper being attempted by slightly less than 8% of the candidates. It was also the least successful scoring, on average, just short of one quarter marks. Four of the five results are given in the question, and many candidates tried to work backwards, albeit in disguised manners. The first results of the question related to SHM. In many cases, candidates did not clearly choose axis or positive directions, and ended with a second order differential equation without a negative sign. It was clear that, in the next part, some did not understand that the particle, being on the platform the whole time, would have the same acceleration as the platform; when writing the equation of motion for the particle, they often included an extra force "from the platform on the particle" equal to mω²(b − x), using the given result. Many also just wrote down the standard equation of motion for SHM, either without having or obtaining a b − x term on the RHS. A few attempted the next section but scored no points. They understood that R ≥ 0 for the platform to remain in contact with the particle, but at no point did they mention the range for x. The last two sections were rarely attempted.